38.5.9 problem 9

Internal problem ID [8357]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.2 Separable equations. Exercises 2.2 at page 53
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 05:29:42 PM
CAS classification : [_separable]

\begin{align*} y \ln \left (x \right ) y^{\prime }&=\frac {\left (y+1\right )^{2}}{x^{2}} \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 22

ode:=y(x)*ln(x)*diff(y(x),x) = (1+y(x))^2/x^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {1}{\operatorname {LambertW}\left (-{\mathrm e}^{\operatorname {Ei}_{1}\left (\ln \left (x \right )\right )-c_1}\right )}-1 \]

✓ Mathematica. Time used: 0.246 (sec). Leaf size: 48

ode=y[x]*Log[x]*D[y[x],x]==( (y[x]+1)/x)^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [\log (\text {$\#$1}+1)-\frac {\text {$\#$1}}{\text {$\#$1}+1}\&\right ]\left [\int _1^x\frac {1}{K[1]^2 \log (K[1])}dK[1]+c_1\right ]\\ y(x)&\to -1 \end{align*}

✓ Sympy. Time used: 0.409 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)*log(x)*Derivative(y(x), x) - (y(x) + 1)**2/x**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = e^{C_{1} + \operatorname {Ei}{\left (- \log {\left (x \right )} \right )} + W\left (- e^{- C_{1} - \operatorname {Ei}{\left (- \log {\left (x \right )} \right )}}\right )} - 1 \]